Girly colours

If Cecilia likes red, according to fact 1, Amy would like pink but considering fact 3, Amy not liking purple means Brenda likes red (and so does Cecilia) and we have two girls liking the same colour which is impossible. Hence red can't be Cecilia's favourite colour. If Cecilia likes purple, based on fact 2, Brenda would like pink, leaving red for Amy but considering fact 3 again, this means Amy doesn't like purple and Brenda would therefore like red which contradicts with our earlier deduction that she likes pink. So purple also isn't Cecilia's favourite colour.

Therefore Cecilia's favourite colour must be pink. Now we know that Amy and Brenda like red and purple but in which order? If Amy likes red (which means Amy's favourite colour is not purple), then Brenda would also like red employing fact 3 and we have two girls liking the same colour which is impossible.

Therefore Amy's favourite colour is purple. Finally, we conclude beyond a reasonable doubt that Brenda's favourite colour is $\boxed{\color{#D61F06}red}$ .

The following propositions hold true :

1) If Cecilia's favourite colour is red, then Amy's favourite colour is pink.

2) If Cecilia's favourite colour is purple, then Brenda's favourite colour is pink.

3) If Amy's favourite colour is not purple, then Brenda's favourite colour is red.

Because none of the above are false, then it must be that for every statement of if P, then Q given, every false Q will also give us a false P while every true P will give us a true Q, too.

Observing statements 2 & 3, at most there should only be one true Q between them, because the fact is Brenda may either like one of the given colours (of either pink or red) OR neither of both (of Brenda liking purple). Because of that, at most there should only be one true P between them, too, as we cannot have a true P and a false Q in a true statement.

Furthermore, the same logic and thought process is also applicable to the pair of the first and second statements. We start with the observation that at most, only one of the two girls likes the pink colour, if it's even them, and for the sole purpose of keeping both statements true, P1 & P2 can't both be true at the same time, the same way that P2 & P3 can't also be both true.

So as P1 <-/-> P2 <-/-> P3 is established, we might be interested in the direct relationship between P1 and P3. Because we cannot have a false statement, we cannot have a true P and a false Q. Therefore, a false Q would force a false P into effect while a true P will also force a true Q to exist. That's why in answering the P1 <--> P3 relationship, we have to consider the interactions between all 4 of them together, of P1, P3, Q1 & Q3.

P1 : C likes red
P3 : A does not like purple
Q1 : A likes pink
Q3 : B likes red

So we can't see many contradictory statements other than the only one pair that can be perceived between P1 and Q3; thus these 2 cannot be both true. We already know that truth is forced on Q sides by P sides, so what we actually have to avoid is P1 and P3 to be both true, since P3 has the power to influence Q3 positively.

Thus, we've already seen that none of the P pairs can be true together, so that can only means one thing : that at most, only one of the Ps can be true alone, if any.

P1 : C likes red
P3 : A does not like purple
P2 : C likes purple
Q1 : A likes pink
Q3 : B likes red
Q2 : B likes pink

Picking up from the previous conclusion that we may only have 1 true P at most but that supposedly true P would force the subsequent Q to also be true, we should make an analysis about the Ps so that a true P will ensure the impossibility of another true P.

P1 & P2 are obviously contradictory that it's crystal clear that one being true will make the other false since they are contesting each other's claim on one same thing, but we can't say the same for a pair of Ps involving P3. Looking at P2 & P3, it's like one is giving up on purpose as to give way to another, and this 'yielding' characteristics does not comply with the original "no 2 true Ps" motto. Thus, P2 must be false so that the 'sacrifice' is rendered useless. Also, between P1 & P3, both have to make sure that the truthfulness that each imposed on their Qs are also contradictory towards the other Ps. Because pink and not-purple is somewhat the same thing (pink is an element in the subset of non-purple colours), that's why P1 must be false or else both P1 & P3 would end up being true.

With the falsities of P1 & P2, C's favourite colour must be pink, and in turn Q1 & Q2 are both false. P3 & Q3 has to have different truth values to take the different colours left, while keeping in mind that the statement as a whole must also be true, which is by taking a false P and a true Q for facts. False P3 makes for A's Purple and true Q3 for B's Red.

Answer : Brenda's favourite colour is Red (with Amy's Purple and Cecilia's Pink).